A Deterministic Distributed 2-Approximation for Weighted Vertex Cover in $$O(\log N\log \varDelta /\log ^2\log \varDelta )$$ Rounds

Ben-Basat, Ran; Even, Guy; Kawarabayashi, Ken-ichi; Schwartzman, Gregory

doi:10.1007/978-3-030-01325-7_21

“…By setting ε = 1/(nW ), we conclude the following result for an f -approximation (recall that we assume that vertex degrees and weights are polynomial in n): Additionally, we get the following range of parameters for which the round complexity is still optimal: For f = O(1) we also get an extension of range of parameters for which the round complexity is optimal. This extension is almost exponential compared to the allowed ε = (log ∆) −O (1) in [BEKS18].…”

Section: Putting It Togethermentioning

confidence: 92%

“…We refer to the additive increase of the dual variable δ(e) as deal(e). Similarly weighted approximation time algorithm to [BEKS18], we use levels to measure the progress made by a vertex. Whenever the level of a vertex increases, it sends a message about it to all incident edges, which multiply (decrease) their deals by 0.5.…”

Section: Tools and Techniquesmentioning

confidence: 99%

Optimal Distributed Covering Algorithms

Basat

¹

,

Even

²

,

Kawarabayashi

³

et al. 2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f . This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f . The approximation factor of our algorithm is (f + ε). Let ∆ denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log ∆/ log log ∆) rounds, for constants ε ∈ (0, 1] and f ∈ N + . This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms.For constant values of f and ε, our algorithm improves over the (f + ε)-approximation algorithm of [KMW06] whose running time is O(log ∆ + log W ), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f -approximation for the problem in O(f log n) rounds, improving over the classical result of [KVY94] that achieves a running time of O(f log 2 n). Finally, for weighted vertex cover (f = 2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [KY11].We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f + ε)-approximaterounds, where f bounds the number of variables in a constraint, ∆ bounds the number of constraints a variable appears in, and M = max {1, ⌈1/a min ⌉}, where a min is the smallest normalized constraint coefficient. This improves over the results of [KMW06] for the integral case, which combined with rounding achieves the same guarantees in O ε −4 · f 4 · log f · log(M · ∆) rounds.In the Minimum Weight Hypergraph Vertex Cover (mwhvc) problem, we are given a hypergraph G = (V, E) with vertex weights w : V → {1, . . . , W }. 1 The goal is to find a minimum weight cover U ⊆ V such that ∀e ∈ E : e ∩ U = ∅. In this paper we develop a distributed approximation algorithm for mwhvc in the congest model. The approximation ratio is f + ε, where f denotes the rank of the hypergraph (i.e., f is an upper on the size of every hyperedge). The mwhvc problem is a generalization of the Minimum Weight Vertex Cover (mwvc) problem (in which f = 2). The mwhvc problem is also equivalent to the Minimum Weight Set Cover Problem (the rank f of the hypergraph corresponds to the maximum frequency of an element). Both of these problems are among the classical NP-hard problems presented in [Kar72].We consider the following distributed setting for the mwhvc problem. The communication network is a bipartite graph H(E ∪ V, {{e, v} | v ∈ e}). We refer to the network vertices as nodes and network edges as links. The nodes of the network are the hypergraph vertices on one side and hyperedges on the other side. There is a network link between vertex v ∈ V and hyperedge e ∈ E iff v ∈ e. The computation is per...

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“…We note that [16] was the first to achieve this running time with no dependence on W , the maximum weight of the nodes. Recently, Ben-Basat et al [6] showed an O(O PT 2 log O PT ) rounds algorithm for computing the minimal (unweighted) vertex cover and a O(O PT ) rounds for a (2 + ε)-approximation. Here, O PT is the size of the smallest cover and thus these algorithms are adequate when a small solution exists.…”

Section: Related Workmentioning

confidence: 99%

Optimal distributed covering algorithms

Ben-Basat

¹

,

Even

²

,

Kawarabayashi

³

et al. 2021

Distrib. Comput.

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2

0

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We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is $$(f+\varepsilon )$$ ( f + ε ) . Let $$\varDelta $$ Δ denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires $$O(\log {\varDelta } / \log \log \varDelta )$$ O ( log Δ / log log Δ ) rounds, for constants $$\varepsilon \in (0,1]$$ ε ∈ ( 0 , 1 ] and $$f\in {\mathbb {N}}^+$$ f ∈ N + . This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and $$\varepsilon $$ ε , our algorithm improves over the $$(f+\varepsilon )$$ ( f + ε ) -approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is $$O(\log \varDelta + \log W)$$ O ( log Δ + log W ) , where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in $$O(f\log n)$$ O ( f log n ) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of $$O(f\log ^2 n)$$ O ( f log 2 n ) . Finally, for weighted vertex cover ($$f=2$$ f = 2 ) our algorithm achieves a deterministic running time of $$O(\log n)$$ O ( log n ) , matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an $$(f\lceil \log _2(M)+1 \rceil +\varepsilon )$$ ( f ⌈ log 2 ( M ) + 1 ⌉ + ε ) -approximate integral solution in $$\begin{aligned} O\left( (1+f/\log n)\cdot \left( {\frac{\log \varDelta }{ \log \log \varDelta } + ({f\cdot \log M})^{1.01}\cdot \log \varepsilon ^{-1}\cdot (\log \varDelta )^{0.01}}\right) \right) \end{aligned}$$ O ( 1 + f / log n ) · log Δ log log Δ + ( f · log M ) 1.01 · log ε - 1 · ( log Δ ) 0.01 rounds, where f bounds the number of variables in a constraint, $$\varDelta $$ Δ bounds the number of constraints a variable appears in, and $$M=\max \left\{ 1, \lceil 1/a_{\min } \rceil \right\} $$ M = max 1 , ⌈ 1 / a min ⌉ , where $$a_{\min }$$ a min is the smallest normalized constraint coefficient.

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“…A maximal matching can be obtained in O(log ∆ + log 3 log n) rounds by plugging the algorithm of Fischer [Fis17] into the framework of Barenboim, Elkin, Pettie, and Schneider [BEPS16]. For weighted minimum vertex cover, the fastest 2-approximation is an O(log n log ∆/ log 2 log ∆) algorithm by Ben-Basat, Even, Kawarabayashi, and Schwartzman [BEKS18], while for a (2 + ǫ)-approximation only the tight O(log ∆/ log log ∆) rounds are needed, as first shown by Bar-Yehuda, Censor-Hillel, and Schwartzman [BCS17].…”

Section: Bhattacharya Et Al [Bchn18mentioning

confidence: 99%

Fast Deterministic Algorithms for Highly-Dynamic Networks

Censor-Hillel,

Dafni,

Kolobov

et al. 2019

Preprint

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This paper provides a simple algorithmic template for obtaining fast distributed algorithms for a highly-dynamic setting, in which arbitrarily many edge changes may occur in each round. Our algorithm significantly improves upon prior work in its combination of (1) having an O(1) amortized time complexity, (2) not posing any restrictions on the dynamic behavior of the environment, (3) being deterministic, (4) having strong guarantees for intermediate solutions, and (5) being applicable for a wide family of tasks which we combinatorially define.The tasks for which we deduce such an algorithm are maximal matching, (degree + 1)coloring, maximal independent set (which, perhaps surprisingly, seems to behave very differently from the other problems), and the seemingly unrelated problem of a 2-approximation for minimum weight vertex cover. For some of these tasks, node insertions can also be among the allowed topology changes, and for some of them also abrupt node deletions.At the core of our work is a combinatorial definition of a subclass of the celebrated family of locally-checkable labelings (LCLs) defined by Naor and Stockmeyer [SIAM J. Comp. '95]. We call the subclass that we define locally-fixable labelings (LFLs). Very roughly speaking, as their name suggests, these are labelings that allow a node to fix its own label and the labels of its neighbors after a topology change, based solely on their old labels.

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A Deterministic Distributed 2-Approximation for Weighted Vertex Cover in $$O(\log N\log \varDelta /\log ^2\log \varDelta )$$ Rounds

Cited by 9 publications

References 23 publications

Optimal Distributed Covering Algorithms

Optimal Distributed Covering Algorithms

Optimal distributed covering algorithms

Fast Deterministic Algorithms for Highly-Dynamic Networks

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